TRIGONOMETRY (MIXED)
R-1
1) Two equal arcs of two circles subtends angle of 60° and 75° at the centre. Find the ratio of the radii of the two circles.
2) If cosθ - sinθ= √2 sinθ show sinθ + cosθ= √2 cosθ
3) If 7 cosθ + 5sinθ= 5, find 5cosθ - 7sinθ.
4) If secθ+ tanθ= x, show that sinθ = (x²-1)/(x²+1).
5) If sinθ + cosecθ= 2 show that sinⁿθ + cosecⁿθ= .
6) If tan⁴θ+ tan²θ= 1, show that cos⁴θ+ cos²θ= 1.
7) If If cis⁴θ+ cos²θ= 1, show that tan⁴θ+ tan²θ= 1.
8) If sinα, cosα, tanα are in GP show that cot⁶α- cot²α= 1.
9) If (secx -1)(secy -1) secx -1)= (secx +1)(secy +1)(secx+1) show the value of each side ± tanx tan y tan z.
10) If tanθ+ sinθ= m and tanθ - sinθ= n show that m²- n²= 4√(mn).
11) If x=cosecα - sinα and y= secα - cosα, show that x²y²(x²+ y²+3)= 1.
12) If cosecα + cosec β + cosecγ = 0, then show (sinα sinβ sinγ)²= sin²α +sin² β + sin²γ.
13) Find the least value of 9 tan²θ + 4 co5²θ.
14) If θ lies in the 2nd quadrant and tanθ = - 5/2 find the value of 2 cosθ/(1- sinθ).
15) If sinθ = 8/15 and sinθ is negative find {sin(θ)+ cos(-θ)}{sec(-θ)+ tan(-θ)}.
16) If α= π/19, show that (sin23 - sin3α)/(sin16α + sin4α)= +1.
17) Evaluate {cot570°+ sin(-330°)/tan(-210°) + cosecx(-750°).
18) If n be any integer; show sin{nπ+(-1)ⁿ π/4}= 1/√2.
19) If α and β are positive acute angls and cos α = 1/√10 and sinβ= 1/√2 find α - β.
20) If x,y are positive acute angles and cosecx =√5, secy = √10/3 find cosecx(x -y).
21) If tan(α +β) + tan(α -β)=4 and tan(α -β) tan(α +β)= 1 find α, β.
R-2
1) Show that sin(α +β)/ sin(α -β)= (tanα tanβ)/(1+ tanα tanβ).
2) Show that cosecx(x +y)= (cosecx cosecx y)/(cotx + cot y).
3) Show that tan40+ tan20=√3(tan45- cot 50 cot70).
4) If tanA + tanB= x and cotA + cotB= y show that cot(A+ B)= 1/x + 1/y.
5) Find the maximum and minimum values of cosθ + √3 sinθ .
6) If tanα = n/(n +1) and tanβ = 1/(2n +1) show that
a) (α +β)= π/4
b) (α +2β)= 1+ 1/n.
7) If tan β)= sin2α/(9+ cos2α) show that 5 tan(α -β)= 4 tanα.
8) If tan²β = tan(α +θ) tan(α - θ ) show that tan² θ = tan(α+ β) tan(α- β).
9) cos²(α- β) + cos²β - 2 cos(α- β) cosα cosβ = sin²α .
10) If cosα + cos(α+ β) + cos(α+ β+γ ) =0 and sinα + sin(α+ β) + sin(α+ β+γ ) =0 show that β = γ =2π/3.
11) If a sin( θ + α)= b sin (θ+ β) show that tanθ = (b sinβ - a cosα)/(a cosα - b cosβ).
12) Simplify: cot(β - γ) cot(γ-α)+ cot(γ -α ) cot(σ -β) + cot(α - β) cot(β- γ).
13) if cos(α -β)/( cos(α +β) + cos(γ+ δ)/cos(γ- δ)= 0 show that tanα tanβ tanγ tanδ= -1.
14) If 9x =π find cosx cos2x cos3x cos4x.
15) Show 4 cos θ cos(60- θ) cos(60+ θ)= cos3θ.
16) Show that sinπ/12 sin3π/12sin5π/12 sin9π/12sin11π/12 = 1/32.
17) Show tan5 tan 55tan65 tan 75= 1
18) 4 sin40 - tan 40= √3.
19) (sin9θ cosθ - cos5θ sin3θ)/(sin16θ cos6θ + cos12θ cos10θ) = tan6θ.
20) If cost = m cosx show that tan{(x - y)/2}= (m -1)/(m +1). cot{(x +y)/2}.
21) If sinθ = n sin(2α- θ) show tan(θ- α) = (n -1)/(n +1) Tan α.
R-3
I) If (1+ m) sin(θ + α)= (1- m) cos(θ - α) show that tan(π/4- θ)= m cot(π/4 - α).
2) If sin(3α+ θ)= 7 sin(α- θ) show tan θ =sinα(1+ sin²α)/{cosα(1+ cos²α).
3) If tan(α -β)= sin2β/{(2n +1) - cos2β)}, show that tanα / tanβ = 1+ 1/n.
4) If cosx + cost + cosx = 0 and sinx + siny + sinx = 0 show that cos{(x +y)/2}= ± 1/2.
5) If cosα + cosβ = -27/65, sinnα + sinβ = 21/65 and π< α -β<3π. Find the values of sin(α +β/2) and cos(α +β/2).
6) If (cosα )/a = (cos(α +β)/b = cos(α +2β)/c = cos(α +3β)/d, then show b(b + d)= c(c + a).
7) Prove: sec2α + tan2α= tan(π/4+ α).
8) Show that cosec50+ √3 sec50= 4.
9) Show: (3- 4 cos2α + cos4α)/(3+ 4 cos2α + cos4α)= tan⁴α.
10) Show that cos(π/7) cos(2π/7)cos(4π/7)= -1/8.
11) If 13 θ= π show that cosθ cos2θ cos3θ cos4θ cos5θ cos6θ = 1/2⁶.
12) If. Tan3A/tanA = k show that sin3A/sinA = 2k/(k -1) and the value of k does not lie between 1/3 and 3.
13) 16 cos⁵θ= cos5θ+ 5 cos3θ+ 10 cosθ.
14) 16 sin⁵θ= sin5θ - 5 sin3θ + 10 sinθ.
15) sin²x cos⁴x = (1/32)(2+ cos2x - 2 cos4x - cos6x).
16) If α and β are positive acute angles and cos2α = (n cos2β -1)/(n - cos2β). Show that √(n -1) tanα = √(n +1) tanβ. (n> 1).
17) If n tanα = (n +1) tanβ, show that tan(α -β)= sin2β/{(2n +1)- cos2β}
18) Show (2cosθ -1)(2 cos2θ -1)(2 cos 2²θ -1) ....(2 cosⁿ⁻¹θ -1)= (2 cos2ⁿθ +1)/(2 cosθ +1).
19) If cos2β = cos(α +γ) sec(α -γ), show that tanα, tanβ, tanγ are in GP.
20) If α and β are two different values of θ lying between 0 and 2π which satisfy the equation 6 cosθ + 8 sinθ = 9, find the value of sin(α + β).
21) Show that value of cot3x/cotx does not lie between 1/3 and 3.
R- 4
1) Show that 2 cosec4θ = sec2θ = (1- tanθ)/(1+ tanθ) cosec2θ.
2) Show that √{(1+ sinθ)/(1- sinθ) = tan(π/4+θ). (0< θ<π/2).
3) Show that cos315/2= (-1/2) √{2+√2}.
4) Show that (sin²24 - sin²6)(sin²42- sin²12)= 116.
5) Show that sin45/4= √{2 - √2+ √2}.
6) Simplify sin(144- x) - sin(144+ x)+ sin(72- x) - sin(72+ x).
7) Show sin(β -γ)+ sin(γ -α) + sin(α -β)+ 4 sin{(β- γ)/2} sin{(γ-α)/2} sin{(α-β)/2} =0.
8) If 270< A< 360 show that 2 sin(A/2)= √(1- sinA) - √(1+ sinA).
9) If α and β are two roots of the equation a cosθ + b sin θ= c show that tan{(α+ β)2}= b/a.
10) Show that sin5 - sin67 + sin 77 - sin139+ sin149= 0.
11) Evaluate cot(15/2)= tan(15/2) - tan(75/2)+ cot(75/2)
12) If cosθ = (cos u - e)/(1- e cosu) show tan(u/2) = √{(1- e)/(1+ e)} tan(θ/2).
13) If A+ B+ C=π. Show that
a) sin²A + sin²B - sin²C = 2 sinA sinB cosC.
b) cotB cotC + cotC cotA + cotA cotB = 1
c) (cotB + cotC)/(tanB+ tanC) + (cotC + cotB)/(tanC + tanA) + cotA+ cotB)/(tanA + tanB)= 1.
d) cosA/(sinB sinC + cosB/sinC sinA + cosC/sinA. SinB = 2
e) cos(A/2) + cos(B/2) + cos(C/2)= 4 cos{(π-A)/4} cos{(π-B)/4} cos{(π- C)/4}.
14) If α +β -γ)=π, show that sin²α + sin²β - sin²γ = 2 sinα sin β cosγ.
15) If A,B,C are the angle of ∆ then show cos²(A/2) - sin²(B/2) - sin²(C/2)= 2 sin(A/2) sin(B/2) sin(C/2).
16) If A+ B+ C= π/2 and cotA, cotB, cotC are in AP show that cotA cotC = 3.
R- 5
Solve:
1) cosmx + cosnx = sin mx + sin mx, m ≠ n.
2) sin(3θ - 30) = cos(2θ+ 10)
3) sin7θ + sin4θ + sinθ = 0, 0≤θ≤π/2.
4) cos3θ + cos2θ= sin(3θ/2) + sin(θ/2), 0≤θ< 2π.
5) sin5x - sin3x - sinx = 0, 0< x <360
6) cot(θ/2) + cosecx(θ/2)= cotθ.
7) 8 cosx cos2x cos4x = (sin6x)/sinx.
8) cotx - 2 sin2x= 1
9) 4 sin2θ cosθ = cpsecθ, 0≤θ≤π.
10) sinx + cosx = 1+ sinx cosx.
11) 3 sinx + 4 cosx = 5.
12) tan²θ + sec2θ= 1.
13) eᶜᵒˢˣ⁺ ˢᶦⁿˣ⁻¹ = 1
14) cosx - sinx = cosθ + sinθ
15) tanx + secx = 2 cosx.
16) tan3θ= tanθ tan(x - θ) tan(x + θ).
17) 2 tan2x + tan3x = tan5x.
18) tan(π/4+ θ) - tan(π/4 - θ)= 2 tanθ tan(π/4 - θ) tan(π/4+ θ)
19) cosx + cos y = 1 and cosx cost= 1/4.
20) If sec ax + sec bx = 0, show that the values of x form two AP.
21) If sinx + sin y =√3(cos y - cosx) show that sin3x + sin3y= 0.
R-6
1) If Uₙ ⁿⁿ₃₅₁₅₇₃⁴²⁴²⁴²⁴²⁴⁴⁸³⁸³³³²²²⁴²⁴²⁴²²²²⁾³⁴³²³²²²²²²²²²²²²²²²²²²²²²²₁²₁²₁²₁₁₁₁₁²²²²⁾³²⁾³²⁾³²⁾³²²²²²²²²²²θφθφᶜᵒˢˣᶜᵒˢˣ θφθφθφαβαβαβαβαβα¹⁾²β⁻¹¹⁾²γ ⁻¹⁻²⁻³⁻¹⁾²αβγθθθθφθφθθθθ θαθβαβαβθθθθθθθθθθθθθθθθθθθθθθαααθθθθθθθθθαβαβαβαβθθθθθθθθθθθθθ²⁾³²⁾³²⁾³ββββββαβαβαγβγαββγβγβ αβγβγαβγθφθφθφαβαβαβαβαβγαβγαβγαβγθθθααθθαβθαβα ααααααααααα θαθθαθααβαβαβθφθφθφθφ
αααθαθθθαα⁻¹θθθθθθθθθθθθθθθθθ⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹ⁿ⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹⁻¹
α φ β σ τ β μ λ γ ⁿ θ δ
Inverse
Simplify:
1) sin(2 tan⁻¹(1/5) - tan⁻¹(5/12).
2) tan(sin⁻¹(1/3) + cos⁻¹(1/√3).
3) sin(2 sin⁻¹(1/√26)+ sin⁻¹(12/13).
4) tan(2tan⁻¹(1/5) - π/4}.
Show
5) 2tan⁻¹2 + tan⁻¹3= π+ tan⁻¹(1/3).
6) 2 cos⁻¹(2/√5)+ cos⁻¹(3/5)= sin⁻¹(24/25).
7) 4 tan⁻¹(1/5) - tan⁻¹(1/239)=π/4
8) cos⁻¹(8/17)+ cos⁻¹(3/5)+ cos⁻¹(36/85)=π.
9) cos⁻¹a - sin⁻¹b = cos⁻¹{b √(1- a²) + a √(1- b²)}
10) tan⁻¹(p/q) - tan⁻¹{(p- q)/(p+ q)}=π/4.
11) tan⁻¹x + tan⁻¹y + tan⁻¹{(1-x - y - xy)/(1+ x + y - xy)}=π/4
12) tan⁻¹{(2a - b)/b√3} + tan⁻¹{(2b - a)/a√3} =π/5
13) cos{(1/3) cos⁻¹(-1/9)}= 2/3.
14) a cos θ + b sin θ = √(a²+ b²) cos(θ - tan⁻¹(b/a) - √(a²+ b²) sin(θ + tan⁻¹(a/b).
15) tan⁻¹{(ap- q)/(aq + p)} + tan⁻¹{(b-a)/(ab +1)} + tan⁻¹{(c- b)/(bc +1)} + tan⁻¹(1/c) = tan⁻¹(p/q).
16) 2 tan⁻¹a + 2 tan⁻¹b = sin⁻¹[{2(a+b)(1- ab)}/{(1+ a²)(1+ b²)}].
Solve
1) tan⁻¹x + tan⁻¹2x + tan⁻¹3x =π.
2) cot⁻¹x + sin⁻¹(1/√5)=π/4.
3) tan⁻¹{(x+1)/(x -2)} + tan⁻¹{(x+1)/(x +2)}=π/4.
4) cos⁻¹(8/x) + cos⁻¹(15/x) =π/2.
5) sin⁻¹x + sin⁻¹(1- x) = sin⁻¹√(1- x²).
6) tan⁻¹{(x +1)/(x -1)} + tan⁻¹{(x -1)/x}= tan⁻¹(-7)., (x ≠ 0,1).
1) If sin⁻¹x+ sin⁻¹y then show that 2(x²- xy + y²)= 1+ x⁴+ y⁴.
2) If tan⁻¹y= 4 tan⁻¹x express y as algebraic function of x.
3) If α = tan⁻¹{(x √3)/(2k - x)} and β = tan⁻¹{(2x k)/k √3} show that one of the value of (α - β) is π/6.
4) If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) =π/2 show that x²+ y²+ z²= r².
COORDINATES AND LOCUS
1) The polar coordinates of the point whose Cartesian coordinates are P(2, -2), are
a) (2√2,π/4) b) (2√2,3π/4) c) (2√2, -3π/4) d) (2√2, -π/4)
2) Rhe polar or coordinates of the point whose Cartesian coordinates are (-√3,1), are
a) (2, 2π/3) b) (2, 5π/6) c) (2, -5π/6) d) (2, -2π/3)
3) The Cartesian coordinates of the point whose polar coordinates are (√3, -3π/4)
a) (√6/2, -√6/2) b) (- √6/2, √6/2) c) (-√6/2, -√6/2) d) none
4) The polar coordinates of the centroid of the Triangle formed by the points A(3,2), B( -6,-3) c) (0,-2) are
a) (√2,π/4) b) (-√2, -π/4) c) (√2, 3π/4) d) (√2, -3π/4)
5) The cartesian coordinates of A and the polar polar coordinates of B are respectively A(2,3) and B(2,60°). The coordinates of the point at which AB is divided internally in the ratio 2:1 are
a) (4/3, √3(2+√3)/3)
b) (5/3, √3(2+√3)/3)
c) (4/3, √3)
d) (5/3, √3)
6) If the coordinates of the centroid of the Triangle formed by the points A(2x, 3x), B(y, 2y), C(-1,-3) are (2,3), then the coordinates of A are
a) (6,6) b) (9,4) c) (4,9) d) none
7) The points (0,-2), (2,4), (-1,-5) form
a) an isosceles triangle b) a right angle triangle c) an equilateral triangle d) none
8) The point (1,2) divides the line segment AB joining the points A (3,2) and B(-2,2) internally in the ratio k: 1. Then k is equals to
a) 2/3 b) 3/2 c) 1/2 d) 2
9) The x-axis divides the line segment AB, where A≡ (2,-3) B≡(5,6), in the ratio
a) 2:3 b) 1:2 c) 2:1 d) 3:2
10) The coordinates of two points A and B are (3,-3) and (-5,7) respective. The line y= x divides AB in the ratio
a) 2:1 b) 1:2 c) 2:3 d) 3:2
11) A(1,6)B(3,-4) and C(x,y) are 3 collinear points such that AB= BC, then the value of x and y satisfy
a) x²+ y²= 220
b) 14y + 5x = 0
c) 13x + 5y+5 = 0 d) none
12) The extremities of the diagonal of a parallelogr are (3,-4) and (-6,5). If the third vertex is the point (-2,1), then the coordinates of the 4th vertex are
a) (-1,0) b) (1,0) c) (0,1) d) (0,-1)
13) The coordinates of the points A, B, C are (-1,-1), (5,7) and (1, -15). The length of the median through A is
a) √41 unit b) √29 unit c) 5 unit d) none
14) The coordinates of the orthocentre of the Triangle, formed by the straight lines given by (x -2)(y -2)(x + y -1)= 0, are
a) (2,-1) b) (- 1,2) c) ( 2,2 ) d) (3/2,3/2).
15) The coordinate of the four points A, B, C, D are (0,0),(0,10),(8,16), (8,6) respectively. If the points are joined in order, then which one is the most appropriate statement ?
a) ABCD forms a square
b) ABCD forms a rhombus
c) ABCD forms a paralellogram
d) none of the statement is true
16) The points (2a,0),(0, 2b) and (1,1) are collinear if
a) 1/a + 1/b = 2
b) 2/a + 2/b = 1
c) 1/a + 1/b = 1 d) none
17) If two vertices of an equilateral triangle have co-ordinates , then for the third vertex which one is most applicable ?
a) the coordinates are integral
b) the coordinates arw rational
c) the coordinates are irrational
d) at least one coordinates is irrational
18) If the coordinates of the midpoints of a triangle ABC are (0,0),(2,-1) and (-1,3), then the coordinations of the centroid of ∆ ABC are
a) (1/3,2/3) b) (-4,-3) c) (-3,-4) d) none
19) The coordinates of the three vertices of a triangle ABC are (-2,1), (-1,-3) and (3,-2); then the coordinates of its circumcentre are
a) (0,0) b) (0,-4) c) (0,-4/3) d) none
20) The equation of the three sides of a triangle are x -2y +4=0, 2x + y - 7 = 0, x + y +3=0. The coordinates of the orthocentre of the triangle are
a) (3/2, 17/12) c) (2,3) c) (3,2) d) none
21) The area of a triangle 5 square unit. Two of the vertices are (2,1) and (3,-2), the third vertex lies on the line y= x +3. The coordinates of the third vertex are
a) (7/2,13/2),(3/2,-3/2)
b) (7/2,13/2),(-3/2,3/2)
c) (-7/2,13/2),(-3/2,3/2) d) none
22) The coordinates of two vertices are (2,2) and (3,1), the third vertex lies on the line y+ 3x = 0. If the coordinates of the centroid of the triangle are (2,0), then the coordinates of the third vertex are
a) (1/3, 1) b) (-1,3) c) (1,-3) d) none
23) The co-ordinates of the the 3 vertices of a triangle are (2,7),(5,1), (x,3) and the area of the triangle 18 sq unit. The values of is/are
a) 10 b) 2,-10 c) 10,-2 d) none
24) The area of the coordinates whose vertices are (a,0), (-B,0), (0,a), (0,-B) (with a,b > 0) is
a) 0 b) (a+ b)²/2 sq unit c) (a²+ b²+ ab)/2 sq unit d) none
25) If the coordinates of the points A, B, C, D are (6,3), (-3,5),( 4,-2) and +x,3x) respectively, and area of ∆ DBC/area of ∆ ABC= 1/2, then the value of x is
a) 8/11 b) 11/8 c) 3/11 d) 11/3
26) The circumcentre of the the triangle formed by the points (-3,1),(1,3) and (3,0) lies on 2x + y= 0. The coordinates of the circumcentre are
a) (1/16,-1/8) b) (-1/8,1/4) c) (-1/16),1/8) d) (1/8,-1/4)
27) A rectangle has two opposite vertices at the points (1,2) and (5,5). If the other vertices lie on the line x = 3, then the coordinates of the other vertices are
a) (3,2),(3,6) b) (3,1), (3,6) c) (3,1),( 3,5) d) (3,-1) ,(3,-6)
28) The locus of the centroid of the triangle whose vertices are (a cosθ, b sinθ), (a sinθ, - b cosθ) and (1,2), where θ is a parameter, is
a) {(3x+1)/a}² + {(3y +2)/b}²= 2
b) (3x-1)² + (3y -2)²= a²- b²
c) (3x+1)² + (3y +2)²= a²- b²
d) {(3x-1)/a}² + {(3y -2)/b}²= 2
29) The locus of the centroid of a triangle whose vertices are (1,0), (a sec θ, a tan θ) and (b sec θ, - b tan θ), where θ is a parameter, is
a) (3x -1)²/(a+ b)² - 9y²/(a - b)²= 1
b) (3x -1)² - 9y²= (a + b)²
c) (3x +1)²/(a+ b)² - 9y²/(a - b)²= 1
d) (3x +1)² - 9y²= a ²- b²
30) If the equation of the locus of a point equidistant from the point (a₁, b₁) and (a₂, b₂) is (a₁ - a₂)x + (b₁ - b₂)y + c= 0, then the value of 'c' is
a) (1/2)(a₁²+ b₁²+ a₂²+ b₂²)
b) (1/2)(a₂²+ b₂² - a₁²- b₁²)
c) (1/2)(a₁² - a₂²+ b₁²- b₂²)
d) √(a₁²+ b₁²- a₁²- b₂²).
31) The locus of the point which divides the line segment joining the points (a,0) and (0,b) in the ratio 2:3 is
a) 2bx - 2ay =0
b) 3bx + 2ay =0
c) 2bx - 3ay =0
d) 2bx + ay =0
32) The locus of the point which divides the line segment joining the points (a cosθ, 0) and (0, b sinθ) in the ratio m: n is
a) x²/a²n² + y²/b²m² = 1/(m + n)²
b) x²/a²m² + y²/b²n² = 1/(m + n)²
c) x²/b²n² + y²/a²m² = 1/(m + n)² d) none
33) A( a cosα , a sinα) and B(b cosβ, b sin β) are 2 points, M(x,y) is another point such that M divides the line segment AB internally in the ratio a: b. Then {tan(α + β)/2} is equals to
a) x/y b) y/x c) (x + y)/(x - y) d) - x/y
34) The locus of the mid-point of the portion of the line x cosθ + y sinθ= p intercepted between the axis is
a) x²+ y²= 4/p² b) x²+ y²= p²/4 c) 1/x²+ 1/y²= p²/4 d) 1/x²+ 1/y²= 4/p²
35) A( a cosα , a sinα) and B(b cosβ, b sin β) are 2 points, M(x,y) is another point such that M divides the line segment AB externally in the ratio a: b. Then {tan(α + β)/2} is equals to
a) x/y b) y/x c) (x + y)/(x - y) d) - x/y
36) The coordinates of the points at a distance 2√2 unit from the point (2,3) in the direction making angle 45° with the positive direction of the x-axis, are
a) (0,1),(4,1) b) (0,5),(4,5) c) (4,5),( 0,1) d) (0,5),(4,1)
37) if the area of the Triangle formed by x cosθ + y sinθ = p with the coordinate axes is always k², then the locus of the midpoint of the segment of the line intercepted between the axes is
a) 2xy = ± k²/p²
b) xy = ± 2k²/p²
c) 2xy = ± k²
d) xy = ± 2k²
38) The parametric coordinates of a point P are {(t+1)/(t -1), 2t+3}, where t is a parameter, then the locus of P is
a) x(y -3)= y -1
b) x(y -5)= y -1
c) x(y -5)= y -3 d) none
39) If θ be a parameter, then the locus of the point P{2/(1+ sinθ), 3 cosθ) is
a) x²y²= 36(x -1) b) x²y²= 36(x +1) c) x²y²+ 36(x -1) = 0 d) x²y²= - 36(x +1)
40) If t be a parameter, then the locus of the point P(2t - 3/t, 2t + 3/t) is
a) x² - y²= 6 b) - x² + y²= 6 c) x² - y²= 24 d) -x² + y²= 24
41) Let P(1,22) and Q(3,4) be two points . The point R on the x-axis is such that PR+ RQ is minimum. The coordinates of R are
a) (3/5,0) b) (5/3, 0) c) (-3/5, 0) d) (-5/3, 0)
42) The Four Points (-a, -B), (0,0),(a, b) and (a², ab) form a
a) parallelogram b) square c) a quadrilateral of area (1/2) ab √(a²+ b²) square unit d) none
43) The base of a triangle lies along the line apx= a and is of length a. If the area of the triangle is a², then its vertex lies on the line
a) x= -2a b) x= 2a c) x= 3a d) x= -3a
44) The locus of the point (2t²+ t +1, t²- t +1) is
a) (x - 2y +1)²= 3(x + y -2)
b) (x - 2y +1)²= 3(x + y +2)
c) (x - 2y +1)²= 3(x - y +2)
d) (x - 2y +1)²= 3(x - y -2)
45) The polar equation of y= x tan with respect to the origin as pole and the+ve y-axis as the initial line is
a) θ=α b) θ= π/2 -α c) θ = -α d) none
46) The cartesian equation of r²= a² cos2θ is
a) (x²- y²)²= a²(x²+ y²)
b) (x²- y²)²/(x²+ y²)= a²
c) (x²- y²)²= a²(x²- y²)
d) (x²- y²)/(x²+ y²)= a²
47) The cartesian equation of θ=α is
a) y = x sinα b) y = x cosα c) y = x tanα d) x = y tanα
48) The Cartesian equation of r cos²(θ/2) =1 is
a) y²= 4(1- x)
b) y²= 4(x - 1)
c) x²= 4(1- y)
d) x²= 4(1+ y)
49) The cartesian equation of √r = √a cos(θ/2) is
a) (2x²+ 2y²+ ax)²= a²(x²+ y²)
b) (2x²+ 2y²- ax)²= a²(x²+ y²)
c) (2x²+ 2y²+ ax)²= a²(x²- y²)
d) (2x²+ 2y²- ax)²= a²(x²- y²)
50) The coordinates of the points A, B, C, P are (6,3),(-3,5),( 4,-2) and (x,y) respectively; then area of ∆ PBC/area of ∆ ABC is equals to
a) |(x + y+2)/7| b) |(x - y+2)/7| c) |(x + y-2)/7| d) none
51) The new coordinates of the point (4,3) when the coordinate axes are translated by shifting the origin to (-2,1) are
a) (6,2) b) (2,4) c) (6,4) d) (2,2)
52) Without changing the direction of the axes, the origin is transferred to the point (-4, -7). The coordinates of a point P in new system, are (5,-2). The coordinates of P in the origin of system, are
a) ( 9,5) b) (1,5) c) ( 9,-9) d) (1,-9)
53) Let P be the image of the point (2,-3) with respect to the x-axis. Then the coordinates of the point P in the new system of coordinates obtained by the translation of axes in which the origin is shifted to (-3,2) are
a) ( 1,1) b) (-1,5) c) (5,1) d) (-1,1)
54) The co-ordinate axes are translated by shifting the origin to the point (-3,4). In the new system of coordinate axes, the respective x and y-intercepts of a straight line l of which the equation in the original system is 2x + 3y= 5, are
a) -1/2, -1/3 b) 11/2,11/3 c) 23/2,23/3 d) none
55) By a translation of Axes if the origin be transferred to (α, β)so that the linear terms in the equation (x + y)(x - y -2)= 4 are eliminated, then the point (α, β) is
a) (1,-1) b) (-1,1) c) (- 1,-1) d) (1,1)
56) If the origin (0,0) is shifted to the point (2,3), by a translation of axes, the equation x²+ y²- 4x - 6y +9=0 changes to
a) x²+ y² + 4 =0 b) x²+ y²- 4=0 c) x²+ y²- 8x - 12y +48=0 d) none
1d 2b 3c 4d 5a 6d 7d 8a 9b 10c 11c 12a 13c 14c 15b 16a 17d 18a 19d 20b 21b 22c 23c 24b 25b 26c 27b 28d 29a 30b 31c 32a 33b 34d 35d 36c 37c 38b 39a 40c 41b 42c 43c 44a 45b 46c 47c 48a 49b 50c 51a 52d 53c 54a 55a 56b
COORDINATES AND STRAIGHT LINES
1) If two vertices of an equilateral triangle have integral co-ordinates then the third vertex will have
a) integral coordinates
b) co-ordinate which are rational
c) at least one co-ordinate irrational
d) coordinates which are irrational.
2) If the line sigment joining (2,3) and (-1,2) is divided internally in the ratio 3:4 by the line x+ 2y= k then k is
a) 41/7 b) 5/7 c) 36/7 d) 31/7
3) The polar coordinates of the vertices of a triangle are (0,0), (3,π/2 and (3,π/6). Then the triangle is
a) right angle b) isosceles c) equilateral d) none
4) The points (a, b+ c),(b + c+ a), (c, a+ b) are
a) vertices of an equilateral triangle
b) collinear c) concyclic d) none
5) The incentre of the triangle formed by the axes and the line x/a + y/b = 1 is
a) (a/2, b/2)
b) {ab/(a+ b +√(ab)),ab/(a+ b +√(ab)}
c) (a/3, b/3)
d) {ab/(a+ b +√(a²+ b²)),ab/(a+ b +√(a²+ b²)}
6) In the ∆ ABC, the coordinates of B are (0,0), AB=2, angle ABC=π/3 and the middle point of BC has the coordinates (2,0). The centroid of the triangle is
a) (1/2,√3/2) b) (5/3, 1/√3) c) ((4+√3)/3, 1/3) d) none
7) The coordinates of these consecutive vertices of a parallelogram (1,3),(- 1,2) and(2,5). The coordinates of the fourth vertex are
a) (6,4) b) (4,6) c) (-2,0) d) none
8) The area of the pentagon whose vertices are (4,1),(3,6),(- 5,1),(-3,3) and (-3,0) is
a) 36 unit² b) 60 unit² c) 120 unit² d) none
9) A point moves in the xy plane such that the sum of its distances from two mutually perpendicular lines is always equal to 3. The area enclosed by the locus of the point is
a) 18 unit² b) 9/2unit² c) 9unit² d) none
10) Let A(=1,2), B=(3,4) and let C=(x,y) be a point such that (x -1)(x -3)+ (y -2)(y -4)= 0. If area (∆ ABC)= 1 then maximum number of positions of C in the xy plane is
a) 2 b) 4 c) 8 d) none
11) The point (α,β),(γ,δ),(α,δ) and (γ,β) taken in order, where α, β, γ, δ are different real numbers are
a) collinear b) vertices of a square c) vertices of a rhombus d) concyclic
12) The diagonals of a parallelogram PQRS are along the lines x+ 3 y= 4 and 6x - 2y =7. Then PQRS must be
a) rectangle b) square c) cyclic quadrilateral d) rhombus
13) The coordinates of the 4 vertices of the quadrilateral are (-2,4),(- 1,2) and (2,4) taken in order. The equation of the line passing through the vertex (-1,2) and dividing the quadrilateral in two equal areas is
a) x+1=0 b)x+y-1=0 c) x - y+3=0 d) none
14) The equation of the straight line which passes through the point (-4,3) such that the portion of the line between the axes is divided internally by the point in the ratio 5:3 is
a) 9x - 20y+ 96=0 b) 9x - 20y -24=0 c) 20x +9y+ 53=0 d) none
15) The equation of the straight line which bisects the intercepts made by the axes on the line x + y =2 and 2x + 3y = 6 is
a) 2x= 3 b) y= 1 c) 2y=3 d) x=1
16) The equation of a straight line passing through the point (-2,3) and making intercepts of equal length on the axes is
a) 2x +y+ 1=0 b) x -y-5=0 c) x -y +5=0 d) none
17) The foot of the perpendicular to the line 3x + y = λ drawn from the origin is C. if the line cuts the x-axis the y-axis at A and B respectively then BC: CA is
a) 1:3 b) 3:1 c) 1:9 d) 9:1
18) The distance of the line 2x - 3y=4 from the point (1,1) in the direction of the line x + y= 1 is
a) √2 b) 5√2 c) 1/√2 d) none
19) The 4 sides of a quadrilaterals are given by the equation xy(x -2)(y -3)=0. The equation of the line parallel to x - 4y=0 that divides the quadrilaterals in two equal areas is
a) x -4y= -5 b) x -4y= 5 c) 4y = x +1 d) 4y+1= x
20) The coordinates of two consecutive vertices A and B of a regular hexagon ABCDEF are (1,0) and (2,0) respectively. The equation of the diagonal CE is
a) √3x + y= 4 b) x + √3 y= -4 c) x + √3 y= 4 d) none
21) ABC is an isosceles triangle in which A is (-1,0), angle A=2π/3, AB= AC and AB is along, the x-axis. If BC=4√3 then the equation of the line BC is
a) x + √3 y= 3 b) √3x + y= 3 c) x + y= √3 d) none
22) The graph of the function cos (x +2) - cos²(x +1) is a
a) straight line passing through the point (0, sin²1) with slope 2
b) straight line passing through the origin
c) parabola with vertex (1, - sin²1)
d) straight line passing through the point (π/2, - sin²1) and parallel to the x-axis .
23) if the points (-2,0),(-1,1/√3) and (cosθ , sinθ ) are collinear then the number of values of θ belongs to [0,2π] is
a) 0 b) 1 c) 2 d) infinite
24) The limiting pisition of the point of intersection of the line 3x + 4y= 1 and (1+c)x + 3c²y= 2 as c tends to 1 is
a) (-5,4) b) (5,- 4) d) (4,-5) d) none
25) The coordinates of the point on the x-axis which is equidistant from the points (-3,4) and (2,5) are
a) (20,0) b) (-23, 0) c) (4/5, 0) d) none
26) The distance between the lines 3x + 4y=9 and 6x + 8y +15= 0 is
a) 3/10 b) 33/10 c) 33/8 d) none
27) If a vertex of an equalateral triangle is the origin and the opposite to it has the equation x + y= 1 then the orthocentre of the triangle is
a) (1/3,1/3) b) (√2/3,√2/3) c) (2/3,2/3) d) none
28) The equation of the three sides of a triangle are x= 2y +1=0 and x+ 2y=4. The coordinates of the circumcentre of the triangle are
a) (4,0) b) (2,-1) c) ( 0,4) d) none
29) L is a variable line such that the algebraic sum of the distances of the points (1,1),(2,0) and (0,2) from the line is equals to zero. The line L will always pass through
a) (1,1) b) (2,1) c) (1,2) d) none
30) ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of the side of the equilateral triangle is
a) 8 b) √(88/3) c) 4√7/√3 d) none
31) If p and p' are the perpendiculars from the origin upon the lines x sec θ+ y cosecθ= a and x cosθ - y sinθ = a cos2θ respectively then
a) 4p²+ p'²= a² b) p²+ 4p'²= a² c) p²+ p'²= a² d) none
32) let the perpendiculars from any point on the line 2x + 11y= 5 upon the lines 24x + 7y= 20 and 4x -3y= 2 have the lengths p and p' respectively. then
a) 2p= p' b) p= p' c) p= 2p' d) none
33) If P(1+ t/√2 , 2+ t/√2) be any points on a line then the range of the values of t for which the point P lies between the parallel lines x+ 2y= 1 and 2x + 4y= 15 is
a) -4√2/5< t< 5√2/6
b) 0< t< 5√2/6
c) 4√2/5< t< 0 d) none
34) There are two parallel lines, one of which has the equation 3x + 4y=2. If the lines cut an intercept of length 5 on the line x+ y= 1 then the equation of the other line is
a) 3x+ 4y= (√6-2)/2
b) 3x+ 4y= (√6+2)/2
c) 3x+ 4y= 7 d) none
35) if the intercept made on the line y= mx by lines y=2 and y= 6 is less than 5 then the range of the value of m is
a) (-∞, -4/3) U (4/3, ∞) b) (-4/3,4/3) c) (-3/4,3/4) d) none
36) If a,b,c are any terms of an AP then the line ax + by + c=0
a) has a fixed direction
b) always passes through a fixed point
c) always cuts intercepts on the axes such that their sum is zero
d) forms a triangle with the axes whose area is constraint
37) If a,b,c are in GP then the line ax+ by + c= 0
a) has a fixed direction
b) always passes through a fixed point
c) forms a triangle with the axes whose area is constant
d) always cuts intercepts on the axes such that their sum is zero
38) The number of the real values of k for which the lines x - 2y +3=0, Kx + 3y +1=0 and 4x - ky +2=0 are concurrent is
a) 0 b) 1 c) 2 d) infinite
39) A family of lines is given by (1- 2k)x + (1- λ)y + λ= 0, λ being the parameter. The line belonging to this family at the maximum distance from the point (1,4) is
a) 4x - y +1=0 b) 33x +12y +7=0 c) 12x +33y -7=0 d) none
40) The members of the family of lines (x + μ)x + (2λ+ μ)y + λ+ 2μ , where λ≠ 0, μ≠ 0, pass through the point
a) (3,1) b) (-3,1) c) (1,1) d) none
41) The equation of of the sides AB, BC and CA of the ∆ ABC are y - x = 2, x+ 2y=1 and 3x + y+5=0 respectively . The equation of the altitude through B is
a) x -3y= -1 b) x -3y= -4 c) 3x - y= -2 d) none
42) The range of values of the ordinate of a point moving on the line x= 1, and always remaining in the interior of the triangle formed by the lines y= x, the x-axis and x + y= 4, is
a) (0,1) b) (0,1] c) [0,4) d) none
43) If the point (a,a) fails between the lines|x + y|= 2 then
a) |a|= 2 b) |a|= 1 c) |a|<1 d) |a|< 1/ 2
44) If A(sinα, 1/√2) and B(1/√2, cosα), -π≤α ≤π, are two points on the same side of the line x - y= 0 then α belongs to the interval
a) (-π/4,π/4)U (π/4,3π/4) b) (-π/4,π/4) c) (π/4,3π/4) d) none
45) The straight line L₁ ≡ 4x - 3y +2= 0, L₂≡ 3x + 4y - 4=0 and L₃≡ x - 7y +6=0
a) form a right angled triangle
b) form a right angled isosceles triangle
c) are concurrent d) none
46) The equation of the bisector of the acute angles between the lines 2x - y +4=0 and x - 2y -1=0 is
a) x +y +5=0 b) x - y +1=0 c) x - y -5=0 d) none
47) The equation of the bisectors of that angle between the lines x +y -3=0 and 2x -y -2=0 which contains the point (1,1) is
a) (√5-2√2)x +(√5+√2)y =3√5- 2√2
b) (√5 +2√2)x +(√5 -√2)y =3√5 +2√2
c) 3x=10 d) none
48) two lines 2x - 3y -1=0 and x +2y +3=0 divide the xy plane in four compartment which are named as shown in the figure. Consider the location of the points (2,-1), (3,2) and (-1,-2). We get
a) (2,-1) belongs to IV
b) (3, 2) belongs III
c) (-1,-2) belongs to II d) none
49) If the lines y- x = 5, 3x+ 4y=1 and y= mx +3 are concurrent then the value of m is
a) 19/5 b) 1 c) 5/19 d) none
50) if the point (cos θ, sinθ) does not fall in that angles between the lines y=|x -1| in which the origin lies then θ belongs to
a) (π/2,3π/2) b) (-π/2,π/2) c) (0,π) d) none
51) The points (-1,1) and (1,-1) are symmetrical about the line
a) y+ x =0 b) y= x c) x+ y=1 d) none
52) The equation of the line segment AB is y= x. If A and B lie on the same side of the line mirror 2x - y= 1, the image of AB has the equation
a) x + y= 2 b) 8x + y= 9 c) 7x - y= 6 d) none
53) Let P= (1,1) and Q=(3,2). The point R on the x-axis such that PR+ RQ is the minimum is
a) (5/3, 0) b) (1/3,0) c) (3,0) d) none
54) If a ray travelling along the line x= 1 gets reflected from the line x + y= 1 then the equation of the line along which the reflected ray travels is
a) y= 0 b) x - y= 1 c) x=0 d) none
55) The point P(2,1) is shifted by 3√2 parallel to the line x + y= 1 in the direction of increasing ordinate , to reach Q. The image of Q by the line x + y= 1 is
a) (5,-2) b) (-1,-2) c) (5,4) d) (-1,4)
56) Let A=(1,0) and B (2,1). The line AB turns about A through an angle π/6 in the clock wise sense, and the new position of B is B'. B' has the coordinates
a) {(3+√3)/2, (√3-1)/2}
b) {(3-√3)/2, (√3+1)/2}
c) {(1-√3)/2, (1+ √3)/2} d) none
57) a line intercepts a, b on the co-ordinate axes. if the axes are rotated about the origin through an angle α then the line has intercepts p, q on the new position of the axes respectively. then
a) 1/p²+ 1/q²= 1/a²+ 1/b²
b) 1/p²- 1/q²= 1/a²- 1/b²
c) 1/p²+ 1/a²= 1/q²+ 1/b² d) none
58) Two points A and B move on the x-axis and the y-axis respectively such that the distance between the two points is always the same. The locus of the middle point of AB is
a) a straight line
b) a pair of straight line
c) a circle d) none
59) Three vertices of a quadrilateral in order are (6,1),(7,2) and (-1,0). If the area of the quadrilateral is 4 unit² then the locus of the fourth vertex has the equation
a) x - 7y= 1 b) x - 7y= -15 c) (x - 7y)²+ 14(x - 7y)- 15 = 0 d) none
60) A variable line through the point (a,b) cuts the axes of reference at A and B respectively. The line through A and B parallel to the y-axis and the x-axis respectively meet at P. Then the locus of P has the equation
a) x/a + y/b = 1 b) x/b + y/a = 1 c) a/x + b/y = 1 d) b/x + a/y = 1
Multiple choice
61) If the coordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational coordinates ?
a) centroid b) inCentre c) circumcenter d) orthocenter
62) Two consecutive vertices of a rectangles of area 10 unit² are (1,3) and (-2,-1). Other two vertices are
a) (-3/5,21/5),(-18/5,1/5)
b) (-3/5,21/5),(-2/5,-11/5)
c) (-2/5,-11/5),(13/5,9/5)
d) (13/5,9/5),(-18/5,1/5)
63) The ends of a diagonal of a square are (2,-3) and (-1,1). Another vertex of the square can be
a) (-3/2,-5/2) b) (5/2,1/2) c) (1/2,5/2) d) none
64) If each of the vertices of a triangle has integral coordinates then the triangle may be
a) right angled b) equilateral c) isosceles d) none
65) if (-1,2),(2,-1) and (3,1) are any three vertices of a parallelogram then the fourth vertex (a,b) will be such that
a) a=2, b=0 b) a=-2, b=0 c) a=-2, b=6 d) a=6, b=-2
66) If (α, β) be an end of a diagonal of a square and the other diagonals has the equation x- y=α then another vertex of the square can be
a) (α - β,α) b) (α,0) c) (0, α) d) (α + β, β)
67) A point on the line y= x whose perpendicular distance from the line x/4 + y/3=1 is 4 has the coordinates
a) (-8/7,-8/7) b) (32/7,32/7) c) (3/2,3/2) d) none
68) The parametric equation of a line is given by
x= -2 + r/√10 and y= 1+3r/√10. then, for the line
a) intercept on the x-axis = 7/3
b) intercept on the y axis = -7
c) slope of the line = tan⁻¹(1/3)
d) slope of the line= tan⁻¹3
69) One side of a square of length a is inclined to the x-axis at an angle α with one of the vertices of the square of the origin. The equation of a diagonal of the square is
a) y( cosα - sinα)= x(cosα + sinα)
b) y( cosα + sinα)= x(cosα - sinα)
c) y (cosα + sinα)- x(cosα - sinα)= a
d) y (cosα -0- sinα)+ x(cosα - sinα)= a
70) If the equation of the hypotenuse and a side of a right angle isosceles triangle be x+ my = 1 and x= k respectively then
a) m= 1 b) m= k c) m= -1 d) m + k= 0
71) The centroid and a vertex of an equilateral triangle are (1,1) and (1,2) respectively. Another vertex of the triangle can be
a) {(2-√3)/2, 1/2}
b) {(2 + 3√3)/2, 1/2}
c) {(2 + √3)/2, 1/2} d) none
72) if one vertex of an equilateral triangles of side 2 is the origin and another vertex lies on the line x= √3 y then the third vertex can be
a) (0,2) b) (√3,-1) c) (0,-2) d) (√3,1)
73) A line passing through the point (2,2) and the exes enclose an area λ. The intercept on the axes made by the line are given by the two roots of
a) x²-2|λ|x + |λ|= 0
b) x² + |λ|x + 2|λ|= 0
c) x²-|λ|x + 2|λ|= 0 d) none
74) A line passes through the origin and making an angle π/4 with the line y - 3x = 5 has the equation
a) x+ 2y=0 b) 2x= y c) x= 2y d) y+ 2x=0
75) The coordinate of a point on the line x+ y = 3 such that the point is at equal distance from the lines |x|=|y| are
a) (3,0) b) (0,3) c) (-3,0) d) (0,-3)
76) A line perpendicular to the line 3x - 2y= 5 cuts off an intercept 3 on the positive side of the x-axis bank. Then
a) the slope of the line is 2/3.
b) the intercept on the y-axis is 2
c) the area of the triangle formed by the line with the axis is 3 unit² d) none
77) One diagonal of a square is the portion of the line √3 x + y= 2√3 intercepted by the axes. Then the extremely of the other diagonal is
a) (1+√3,√3-1) b) (1+√3,√3+1) c) (1-√3,√3-1) d) (1-√3,√3+1)
78) if bx + cy = a, where a, b, c are of the same sign , be a such line that the area enclosed by the line and the axes of reference is 1/8 unit² then
a) b,a,c are in GP
b) b, 2a, c are in GP
c) b, a/2, c are in AP
d) b, -2a, c are in GP
79) The side of the triangle are x + y = 1, 7y= x and √3 y + x=0. then the following is an interior point of the triangle.
a) circumcenter b) centroid c) incentre d) orthocenter
80) If (x, y) be a variable points on the line line y= 2x lying between the lines 2(x +1)+ y= 0 and x + 3(y -1)=0 then
a) x∈ (-1/2,6/7) b) x∈ (-1/2,3/7) c) y∈ (-1,3/7) d) y∈ (-1,6/7)
81) if the equation of the three sides of a triangle are x+ y= 1, 3x + 5y=2 and x - y=0 then the orthocentre of the triangle lies on the line
a) 5x - 3y= 2
b) 3x - 5y= -1
c) 2x - 3y= -1
d) 5x - 3y= 1
82) A ray travelling along the line 3x - 4y= 5 after being reflected from a line l travels along the line 5x + 12y= 13. then the equation of the line l is
a) x +8y= 0 b) x = 8y c) 52x + 4y= 65 d) 32x - 4y= -65
83) A ray of light travelling along the line x +y= 1 is incident on the x-axis and after refraction it enters the other side of the x-axis by turning π/6 away from the x-axis. The equation of the line along which the refracted ray travels is
a) x +(2- √3)y= 1 b) (2- √3)x +y= 1 c) (2+√3)x = 2+√3 d) none
1c 2a 3c 4b 5d 6b 7b 8a 9a 10b 11b 12d 13c 14a 15b 16c 17d 18a 19a 20c 21a 22d 23b 24a 25d 26b 27a 28a 29a 30c 31a 32b 33a 34d 35a 36b 37c 38a 39c 40a 41b 42a 43c 44a 45c 46b 47a 48a 49c 50b 51b 52c 53a 54a 55d 56a 57a 58c 59c 60c 61acd 62 ac 63 ab 64 acd 65 bd 66 bd 67 ab 68 d 69 ac 70 ac 71 ac 72 ab 73c 74cd 75ab 76 bc 77 bc 78 bd 79 bc 80 bd 81 bd 82 bc 83 ac
α θ β γ σ φ ω μ λ ₁₂₃₄₅₆₇₈₉₀ ∞
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point distance from the points 20211 to a variable straight lines be zero then the line if the vertices of the triangle obtained by the joining the midpoints of the sides of the triangle 0 to 20 13 then the triangle is with vertices and the sum of the abscissa of all points on the line that lie at a unit distance from the line on the position of the straight line which is intersected between the square is constructed away from the origin with the person has one of its side if you do not the perpendicular distance of the square from the origin and the maximum value of is the lion made sex Axis at and y axis at is the mid point of is a foot of the perpendicular from the line mids the excess of X and Y at respectively a triangle is inscribed in a triangle being the origin with the right angle at and light temporary on area of the triangle of the area of the triangle if P are the length of the perpendicular from the origin to the straight lines equations are the base of a triangle is 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the point of intersection of the line is constant there are two straight line passing through the point 273 units between the lines supposed the lines not interested any point in the plane the earth the centre of the triangle with the vertices concert with the middle point of the one of the sides the point 41 undergoes the following transformation successfully reflection about the line translation through a distance to units along the positive direction of x-axis rotation through an angle about the origin in anticlockwise direction reflection about the final position of the given point is find the Lok Sabha point is moves so that the sum of its distance from the points are in points on a plane of coordinates are respectively biseptic at the point is divided the ratio is divided the ratio 1:3 is divided the ratio 1:4 and 10 points in exhausted so that the coordinate of the final points so obtained hour a line interested the three sides of a triangle respectively so that so that the area of the triangle the vertices is independent the circumstance of the triangle of the vertices lies at the origin where so that's it for to centralised on the line prove that no line can be drawn through the point 45 so that is distance from 23 will be positive so that the reflection of the line in the line is the line where a triangle as the lines as to a presides with being a routes of the equation is a to centre of the triangle so that equation of third side is equation of the days of an equilateral triangle and other vertex of the point 12 find the equation of the order side and the length of the side of the triangle prove that all lines represent by the equation pass through it fixed point for all values of find the coordinates of this point and replaced in the line ardhak Shabd dekhoge sanshodhan D area D Triangle formed by the lines the sides of a rhombus ABCD are parallel lines and if the diagonals of the Rhombus intersect at the point 12 and the vertex is on the y-axis find the possible coordinates a line which makes an acute angle with the positive direction of excess join through the point 34 the cut the car so that the lens of the sequence are numerical value of the roots of the Coorg a triangle ABC right angle at C with C is equal to b more such that the angular points slide along the y axis respectively find the locals of 2 points are given is a variable point on the side of the find the locus of from the FB with the common difference respectively so that the point where tense of the base of an isosceles Triangles are equation of one side is find the equation of the base the remaining side and the coordinates of the vertex straight line is such that everything and the perpendicular falling upon A from any number of fixed point is zero so that its always passes through a fixed point equation of the bisector of the angle between the lines three lines on the three sides of the squares find the equation of the four side of the squares square lies about takes of the original one of the size passing the origin makes an angle with the positive direction that is prove that the equation of the diagonals are where is the length of the square find equations 13 and 24 straight line drawn through the point 21 is such that its points of intersection with the line at a distance 32 from the point find the direction of the line the points 1234 246 of the triangle the point 38 lies on produced find the coordinate of the coordinates of the midpoint of the sides of the triangle are 2153 and 37 find the equation of the sides of the triangle 360 are 250 points is a variable points of the plane mirrors access respectively and its prove that passes through 20 for any position in the place find the condition of the centre of the Triangle formed by the line and show that value that the central lies on the line the coordinate of the point satisfied the relation so that the centre of the triangle are satisfy the relation the original straight line is a drone to cut the lines respect to be find the locus of the point on the triangle lines such that is the geometric mean between if the vertex of a triangle have a integral coordinates prove that it cannot be collateral or vertices of a triangle so that the equation through is given by
PERMUTATION
1) How many odd numbers of 6 digits can be formed with the digits 0, 1, 4, 5, 6, 7, none of the digit being repeated in any numbers ? 288
2) From 10 different 6 things are taken at a time so that a particular thing is always included. Find the number of such permutations . 90720
3) How many different arrangements of the letters of the word FAILURE can be made so that the vowels are always together ? 576
4) In how many ways can the letters of the word STATION be arranged so that the vowels are always together ? 360
5) How many words can be formed by the letters of the word PEOPLE taken all together so that the two Ps are not together ? 120
6) In how many ways 6 books be arranged on a shelf of an almirah so that 2 prticular books will not be together? 480
7) Find the values of n and r when ⁿPᵣ = 336 and ⁿCᵣ = 56. 8,3
8) A box contains 10 electric lamps of which 4 are defective . Find the number samples of 6 lanpa taken at random from the box which will contain two detective lamps. 90
9) A committee of five is to be formed from 6 boys and 4 girls. How many different committees can be formed so that each committee contains at least two girls ? 186
10) An examinee has to answer 6 questions out of 12 questions . The questions are divided into two groups , each group containing 6 questions. The examinwe is not permitted to answer more than 4 questions from any group. In how many ways can he answer in all 6 questions ? 850
11) The Indian cricket eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that team contains at least three bowlers? 1260
12) In a plane 5 points out of 12 points are collinear, no three of the remaining points are callinear. Find the number of straight lines formed by joining these points. 57.
13) Find the values of n and r, when ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁. 3,2
14) which polygon has the same number of diagonals as sides ? Pentagon
15) How many words can be made out of the letters of the word EQUATION taken all at a time such that each word will have one constant in the beginning and one at the end. 4320
16) How many numbers of 7 digits can be formed with the digit 1,2, 3,4, 3, 2, 1 so that the odd digits will occupy odd places. 18
17) How many numbers greater than 1 lakh can be formed with the digits 0, 2, 5, 2, 4, 5 ? 150
18) In how many telephone numbers of 6 digits, two consecutive digits will be different? 9⁶
19) m men and n women take seats in a row. If m > n and no two women sit together then show that they can take seats in m!(m+1)!/(m - n +1)! ways.
20) Two person A and B sit with other 10 persons in a row. In how many arrangements will there be three persons in between A and B ? 16. 10!
21) Find the rank of the word MOTHER when its letters are arrange as in a dictionary. 309
22) 18 guests have to be seated, half on each side of a long table. 4 particular guests desire to sit on one particular side and three others on other side. Determine the number of ways in which the sitting arrangementa can be made. 2 x ¹¹C₅ x (9!)²
23) Show that the total number of permutations of n different things taking and more than r things at a time (repetition is allowed ) is n(nʳ -1)/(n -1).
24) If ⁿCᵣ₋₁/a = ⁿCᵣ/b = ⁿCᵣ₊₁/c, then show that,
n= (ab + 2ac + bc)/(b²- ac) and r= a(c + b)/(b²- ac)
25) If ⁿ⁺¹Cₘ₊₁ : ⁿ⁺¹Cₘ : ⁿ⁺¹Cₘ₋₁ = 5:5:3, find the values of m and n. 3,6
26) ⁴ⁿC₂ₙ/²ⁿCₙ = {1.3.5....(4n -1)}/{1.3.5....(2n -1)²}.
27) A committee of five persons is to be formed from 6 ladies and 4 gentlemen. How many committees containing at least two ladies can be formed where Mr A and Mrs B will
a) always remain
b) never remain. 55,56
28) Different arrangements are made by taking 3 vowels and 5 consonants out of 5 vowels and 10 consonants respectively so that the vowels always come together. Find the number of such arrangements. 10886400
29) In how many ways 4 or more men from 10 men can be selected ? 848
30) A box contains two white balls , 3 black balls and 4 red balls. In how many ways can three balls be drawn from the box if atleast one black ball is to be included in the draw ? 64
31) Find the total number of combinations taking at least one green ball and one blue ball from 5 different green balls, 4 different blue balls and 3 different red balls. 3720
32) A student is allowed to select atmost n books from a collection of (2n +1) books. If the total number of selections of at least one book is 63, then find the value of n. 3
33) A student has to select even number of books from a collection of 2n books. if he can select books in 2047 different ways, find the value of n. 6
34) How many different selections can be done taking at least one letter from each of the words TABLES , CHAIR, BENCH (2⁵-1)(2⁵-1)2⁵-1)= 29791
35) 2n out of 3n articles are alike and the rest are different. In how many ways 2n articles out of 3n articles can be selected. 2ⁿ
36) A box contains 2 guavas , 3 orange ps, 4 apples of different shapes.
a) in how many ways one or more fruits can be selected ?
b) how many selections of fruits can be made taking at least one of each kind? 511, 315
37) A question paper contains 10 questions . Four answers for each question are given of which one is correct. If one examinee gives answers for all of the 10 questions and select one answers for each question then in how many ways he can give correct answers for 5 questions? 61236
38) If any 7 dates are named at random then in how many cases of them, there will be three Sundays . 45360
39) There are two women participating in a chess competition. Every participant played 2 games with other participants. The number of games the men played between themselves exceed by 66 the number the meen played with the women . How many participated in the tournament and how many games were played? 13, 156
40) 5 balls of different colours that we placed in three boxes of different sizes All the five balls can be placed in each box. In how many different ways the balls can be kept in the boxes so that no box will be empty ? 150
41) In how many ways 12 different fruits can be divided among three boys so that each one can get at least one fruit ? 519156
42) There are four balls of four colours and 4 boxes of same colours with the balls. In how many different ways each box will contain one ball so that no ball will go to the box if its own colour. 9
43) a box contain 5 pair of shoes. In how many different ways can 4 shoes be selected so that there will be no complete pair of shoes? 80
44) A, B, C have 5,3 and 7 books of different types respectively. In how many different ways can they exchange the books so that number of books of everyone remains as before? 15!/(5!3!7!)
45) How many a) selections and b) arrangement can be made taking 4 letters from the word PROPORTION ? 53,758
46) How many numbers of 4 digit can be formed with the digits 1,1,2, 2, 3, 3, 4, 5 ? 354
47) Find the number of arrangements taking 5 letters at a time from the letters a,a,a,b,b,b,c,d. 320
48) How many different arrangements can be made by taking Four Pens from 3 same red colour pens, 2 same blue colours pens and 3 pens of other different colour ? 286
49) No 3 diagonals of a decagon are concurrent except at the vertices . Find the number of points of interaction of the diagonals. 595
50) AB and CD are two parallel straight lines. 20 points on AB and 20 points in CD are taken and the points on AB are connected with the points on CD . if no two straight lines are parallel and no three points are concurrent then how many Triangles can be formed so that one of the angular points of each triangle lies on AB and one on CD and another vertex does not lie on AB or CD? 31920000
51) How many Triangles can be obtained by joining the vertices of a polygon of 24 sides so that the sides of the triangle will not be the sides of the polygon ? 1520
52) The length and breath of a parallelograms are cut by p lines being parallel to both length and breadth . Show that in all (1/4)(p+1)²(p+2)² parallelograms are formed.
53) In a place there are m parallel roads along north-south and n parallel roads along east -west. In how many different short routes can a man go from the junction of north- east to the junction of Southwest? ⁿ⁺ᵐ⁻²Cₙ₋₁ (m+ n -2)!/{(n -1)!(m-1)!}
54) Find the number of different squares those can be formed on a chessboard. 204
55) There are n letters and n directed envelopes. In how many ways could all the letters be put into wrong envelopes ? n! (1/2! - 1/3! + 1/4! - .....+ (-1)ⁿ/n!)
56) In how many ways can 3A's, 2B's and 1C's be arranged in one line so that 2A's never occur together? 12
57) The number of 5 digit telephone numbers , none of their digits being repeated is
a) 50 b) ⁵⁰P₅ c) 5¹⁰ d) 10⁵
58) The number of 10 digit numbers formed with the digit 1 and 2, is
a) ¹⁰C₁ + ⁹C₂ b) 2¹⁰ c) ¹⁰C₂ d) 10!
59) A, B, C, D and E have been asked to deliver a lecture in a meeting. In how many ways can their lectures be arranged so that C delivers lecture just before A? 24
60) How many different signals can be given by using any number of the flags from six flags of different colours ? 1956
61) The number of ways in which five unlike rings a man can be wear on the four fingers of one hand
a) 120 b) 625 c) 1024 d) none
62) Show that the product of r successive natural number is divisible by r!.
63) If n> 7, prove that ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₂.
64) The value of ⁴⁷C₄ + ⁵ᵢ₌₁∑⁵²⁻ⁱC₃ is
a) ⁴⁷C₄ b) ⁵²C₃ c) ⁵²C₄ d) none
65) The value of ¹⁵C₁+ ¹⁵C₃ + ¹⁵C₅+....+ ¹⁵C₁₅ =
a) 15!16! b) 15.2⁸ c) 2¹⁴ d) 2¹⁵
66) In a football championship, there were played 153 matches. Every two teams playex one match with each other. The number of teams, participating in the championship is
a) 17 b) 18 c) 9 d) none
67) Everybody in a room shakes handa with everybody else. The total number of hand shakes is 66. The total number of persons in the room is
a) 11 b) 12 c) 13 d) 14
68) The number of ways to form a team of 11 players out of 22 players where 2 particular players are included and 4 particular players are never included in the team is
a) ¹⁶C₁₁ b) ¹⁶C₅ c) ¹⁶C₉ d) ²⁰C₉
69) The total number of factors of 1998 (including 1 and 1998), is
a) 18 b) 16 c) 12 d) 10
70) In how many ways can 9 different things be divided into 3 groups of 2, 3 and 4 things respectively ? 1260
71) In how many ways can 12 different things be divided equally into 4 groups ? 15400
72) Of the four numbers 25, 150 , 170, 210 which one is the number of diagonals of a polygon of 20 sides ? 170
73) If a polygon has 54 diagonals, find the number of sides of the polygon. 12
74) In how many ways can the result (win or loss, or draw) of 3 successive football matches be decided? 27
75) There are 10 electric bulbs in a hall. Each of them can be lightened separately. The number of ways for lightning the ball is
a) 10² b) 1023 c) 2¹⁰ d) 10!
76) How many quadrilaterals can be formed with the seven sides of the length 1cm, 2 cms, 3 cms, 4 cms, 5 cms, 6 cms and 7 cms ? 32
77) How many different algebraic expression can be formed by combining the letters a, b, c, d, e, f with the signs '+' and '-', all the letters taken together? 64
78) Find the total number of ways in which six '+' and four '-' signs can be arranged in a line such that no two '-' signs occur together. 35
1) If ⁿCᵣ₋₁ = 56, ⁿCᵣ = 28 and ⁿCᵣ₊₁ = 8 then r is equal to
a) 8 b) 6 c) 5 d) none
2) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀ ∑ ⁴⁹⁺ʲC₁₀₊ⱼ is equal to
a) ⁵¹C₂₀ b) 2. ⁵⁰C₂₀ c) 2. ⁴⁵C₁₅ d) none
3) In a group of boys, the number of arrangements of 4 boys is 12 times the number of arrangements of 2 boys. The number of boys in the group is
a) 10 b) 8 c) 6 d) none
4) The value of ¹⁰∑ᵣ₌₁ r. ʳPᵣ is
a) ¹¹P₁₁ b) ¹¹P₁₁ -1 c) ¹¹P₁₁ +1 d) none
5) From a group of persons the number of ways of selection 5 persons is equals to that of 8 persons . The number of person in the group is
a) 13 b) 40 c) 18 d) 21
6) The number of distinct rational numbers x such that 0< x <1 and x= p/q, where p, q belongs to {1,2,3,4,5,6} is
a) 15 b) 13 c) 12 d) 11
7) The total number of 9 digit numbers of different digits is
a) 10. 9! b) 8.9! c) 9. 9! d) none
8) The number of 6 digit numbers that can be made with the digits 0, 1, 2, 3, 4 and 5 so that even digits occupy odd places is
a) 24 b) 36 c) 48 d) none
9) The number of ways in which 6 men can be arranged in a row so that 3 perpendicular men are consecutive, is
a) ⁴P₄ b) ⁴P₄ x ³P₃ c) ³P₃ x ³P₃ d) none
10) Seven different lecturers are to deliver lectures in 7 periods of a class on a particular day. A, B and C are three of the lecturers . The number of ways in which a routine for the day in the can be made such that A delivers his lectures before B, and B before C, is
a) 420 b) 120 c) 210 d) none
11) The total number of 5 digit numbers of different digits in which the digit in the middle is the largest is
a) ⁹ₙ₌₄∑ⁿP₄ b) 33(3!) c) 30. 3! d) none
12) A five digit number divisible by 3 to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is
a) 216 b) 600 c) 240 d) 3125
13) let A=x : x is a prime number and x< 30}. The number of different rational numbers whose numerator and denominator belongs to A is
a) 90 b) 180 c) 91 d) none
14) The total number of ways in which 6 '+' and 4 '-' signs can be arranged in a line such that no two '-' signs occur together is
a) 7!/3! b) 6! 7/3/ c) 35 d) none
15) The total number of words that can be made by writing the letters of the word PARAMETER so that no vowels is between two consonants is
a) 1440 b) 1800 c) 2160 e) none
16) The number of numbers are four different digits that can be formed from the digit of the number 1 2 3 5 6 such that the numbers are divisible by 4, is
a) 36 b) 48 c) 12 d) 24
17) let S be the set of all fractions from the set A to the set A. If n(A)= k then n(S) is
a) k! b) kᵏ c) 2ᵏ -1 d) 2ᵏ
18) Let A be the set of the four digit numbers a₁a₂a₃a₄ where a₁ >a₂>a₃>a₄ then n(A) is equal to
a) 126 b) 84 c) 210 d) none
19) The number of numbers divisible by 3 that can be formed by 4 different even digits is
a) 18 b) 36 c) 0 d) none
20) The number of 5 digit even numbers that can be made with the digit 0, 1 , 2 and 3
a) 384 b) 192 c) 768 d) none
21) The number of 4 digit number that can be made with the digit 1, 2, 3, 4 and 5 in which at least two digits are identical is
a) 4⁵ - 5! b) 505 c) 600 d) none
22) the number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant is
a) 5.7!/2 b) 3.7!/2 c) 2.7! d) none
23) The number of numbers of 9 different digits such that all the digits in the first four places are less than the digits in the middle and all the digits in the last four places are greater than that in the middle is 2.4! b) (4!)² c) 8! d) none
24) In the decimal system of numeration the number of 6 digit numbers in which the digits in any place is greater than the digit to the left of it is
a) 210 b) 84 c) 126 d) none
25) the number of 5 digit numbers in which no 2 consecutive digits are identical is
a) 9² . 8³ b) 9x 8⁴ c) 9⁵ d) none
26) in the decimal system of numeration the number of 6 digit numbers in which the sum of the digits is divisible by 5 is
a) 180000 b) 540000 c) 5 x 10⁵ d) none
27) The sum of all the numbers of four different digits that can be made by the using 0, 1, 2,and 3: is
a) 26664 b) 39996 c) 38664 d) none
28) A teacher take three children from her class to the zoo at time as often as she can, but she does not take the same three childrens to the zoo more than once. she finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is
a) 12 b) 10 c) 60 d) none
29) ABCD is a convex quadrilateral 3, 4, 5 and 6 points are marked on the sides AB , BC, CD and DA respectively. The number of triangles with vertices on different sides is
a) 270 b) 220 c) 282 d) none
30) there are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points is
a) 116 b) 120 c) 117 d) none
31) in a polygon the number of diagonals is 54. The number of sides of the polygon is
a) 10 b) 12 c) 9 d) none
32) in a polygon no 3 diagonal concurrent. If the total number of points of intersection of the diagonals interior to the polygon be 70 then the number of diagonals of the polygon is
a) 20 b) 28 c) 8 d) none
33) n lines are drawn in a plane such that no two of them are parallel and no 3 of them are concurrent. The number of different points at which these lines will cuts is
a) ⁿ⁻¹∑ₖ₌₁ k b) n(n -1) c) n² d) none
34) The number of triangles that can be formed with 10 points as vertices, n of them being collinear , is 110. Then n is
a) 3, 4, 5, 6
35) There are three coplanar parallel lines. if any p points are taken on each of the lines, the maximum number of triangles with vertices at these points is
a) 3p²(p -1)+ 1 b) 3p²(p -1) c) p²(4p -3) d) none
36) Two teams are to play a series of 5 matches between them. A match ends in a way or loss or draw for a team. A number of peoples forecast the result of each match and no two people make the same forecast for the series of the matches. The smallest group of people in which one person forecast correctly for all the matches will contain n people, where n is
a) 81 b) 243 c) 486 d) none
37) A bag contain three black, 4 white and two red balls, all the balls bings different. The number of selections of atmost six balls containing balls of all the colours is
a) 42 (4!) b) 2⁶ x 4! c) (2⁶-1)(4!) d) none
38) in a room there are 12 bulbs of the same wattage, each having a separate switch . The number of ways to light the room with different amounts of illumination is
a) 12²-1 b) 2¹² c) 2¹²-1 d) none
39) In an examination 9 papers of candidates has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can unsuccessful is
a) 255 b) 256 c) 193 d) 319
40) The number of 5 digit numbers that can be made using the digit 1 and 2 and in which at least one digit is different, is
a) 30 b) 31 c) 32 d) none
41) In a club election the number contestants is one more than the number of maximum candidates for which a voter can vote . if the total number of ways in which a voter can vote be 62 then the number of candidates is
a) 7 b) 5 c) 6 d) none
42) The total number of selection of atmost n things from (2n +1) different things is 63. Then the value of n is
a) 3 b) 2 c) 4 d) none
43) let 1≤ m< n ≤ p. The number of subsets of the set A={1,2,3,....p} having m,n as the least and the greatest elements respectively, is
a) 2ⁿ⁻ᵐ⁻¹ -1 b) 2ⁿ⁻ᵐ⁻¹ c) 2ⁿ⁻ᵐ
44) The number of ways in which n different prizes can be distributed among m(<n) persons if each is entitled to receive atmost n -1 prizes ,is
a) nᵐ b) mⁿ c) mn d) none
45) The number of possible outcomes in a throw of n ordinary dies in which at least one of the dice shows an odd number is
a) 6ⁿ-1 b) 3ⁿ -1 c) 6ⁿ - 3ⁿ d) none
46) The number of different 6 digit numbers that can be formed using the three digit 0, 1, 2 is
a) 3⁶ b) 2x 3⁵ c) 3⁵ d) none
47) The number of different matrices that can be formedd with elements 0, 1, 2norn3, each Matrix are 4 element, is
a) 3x 2⁴ b) 2x 4⁴ c) 3 x 4⁴ d) none
48) Let A be a set of n(≥ 3) distinct elements . The number of triplets (x,y,z) of the elements of A in which at least two co-ordinates are equal is
a) ⁿP₃ b) n³ - ⁿP₃ c) 3n² - 2n d) 3²(n -1)
49) The number of different pairs of words (_ _ _, _ _ _) that can be made with the letters of the word STATICS is
a) 828 b) 1260 c) 396 d) none
50) Total number of 6 digit numbers in which all the odd digits and only odd digit appear, is
a) 5(6!)/2 b) 6! c) 6!/2 d) none
51) The number of divisors form 4n +2(n≥ 0) of the integer 240 is
a) 4 b) 8 c) 10 d) 3
52) in the next World Cup of cricket there will be 12 teams , divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top team will qualify for the next round. In this round each team will play against other once. Four top teams of this round will qualify for the semi final round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of 3 matches. The minimum number of matches in the next world cup will be
a) 54 b) 53 c) 38 d) none
53) The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is
a) 60(5!) b) 15(4!(5!) c) 4! x 5! d) none
54) four couples (husband and wife) decide to form a committee of four members . The number of different committees that can be formed in which no couple finds a place is
a) 10 b) 12 c) 14 d) 16
55) from four gentlemen and 6 ladies a community of 5 to be selected. The number of ways in which the committee can be formed so that is gentlemen are in majority is
a) 66 b) 156 c) 60 d) none
56) There are 20 questions in a question paper. If no two students solve the same combination questions but solve equal number of questions then then the maximum number of student who appeared who appeared in the examination is
a)²⁰C₉ b) ²⁰C₁₁ c)!²⁰C₁₀ d) none
57) 9 hundred distinct n-digit positive numbers are to formed using only the digits 2, 5 and 7. The smallest value of n for which this is possible is
a) 6 b) 7 c) 8 d) 9
58) The total number of integral solutions (x, y, z) such that XYZ= 24 is
a) 36 b) 90 c) 120 d) none
59) The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied by consonants is
a) 576 b) ⁴C₃. 4! c) 2(4!) d) none
60) a cabinet ministers consist of 11 ministers, one ministers being the chief minister. A meeting is to be held in a room having a round table and 11 chairs round it, one of them being meant for the chairman. The number of ways in which the ministers can take their chairs, the chief minister occupying the chairman's place is
a) 10!/2 b) 9! c) 10! d) none
61) The number of a ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats is
a) 1440 b) 720 c) 5040 d) none
62) The number of ways in which 20 different pearls of two colours can be set alternatively on a necklace there, being 10 pearls of each colour, is
a) 9! x 10! b) 5(9!)² c) (9!)² d) none
63) If r> p > q, the number of different selections of p+ q things taking r at a time, where p things are identical and q things are identical is
a) p+ q+ r b) p+ q- r +1 c) - p- q+ r +1 d) none
64) There are four mangoes , three apples, two oranges and one each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of each kind is
a) 10! b) 9! c) 4! d) none
65) The number of proper divisors of 2ᵖ. 6ᑫ. 15ʳ is
a) (p+ q+1)(q+r+1)(r+1)
b) (p+ q+1)(q+r+1)(r+1) -2
c) (p+ q)(q+r)r -2
d) none
66) The number of proper divisors of 1800 which are divisible by 10, is
a) 18 b) 34 c) 27 d) none
67) The number of odd proper devisors of 3ᵖ. 6ᑫ. 21ʳ is
a) (p+1)(m+1)(n+1)-2
b) (p+ m+ n+1)(n+1) -1
c) (p+ 1)(m+1)(n+ 1)
d) none
68) The number of even proper devisors of 1008 is
a) 23 b) 24 c) 22 d) none
69) in a test there were n questions. In the test 2ⁿ⁻ⁱ students of wrong answers to i questions where i= 1, 2, 3,.....n. if the total number of wrong answers given is 2047 then n is
a) 12 b) 11 c) 10 d) none
70) The number of ways to give 16 different things to 3 persons A, B and C so that B gets one more than A and C gets two more than B, is
a) 16!/(4!5!7!)
b) 4!5!7!
c) 16/(3! 5!8!) d) none
71) The number of ways to distribute 32 different things equally among 4 persons is
a) 32/(8!)³ b) 32!/( 8!)⁴ c) 32!/4 d) none
72) if 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide 3n things in 3 equal groups is
a) k x 3 b) k/3! c) (3!)ᵏ d) none
73) in a packet there are m different books, n different pens and p different pencils. The number of selections of at least one articles of each type from the packet is
a) 2ᵐ⁺ⁿ⁺ᵖ -1 b) (m+1)(n +1)(p+1) -1 c) 2ᵐ⁺ⁿ⁺ᵖ d) none
74) The number of 6 digits number that can be made with the digit 1, 2, 3 and 4 and having exactly two pairs of digit is
a) 480 b) 540 c) 1080 d) none
75) The number of words of 4 letters containing equal number of vowels and consonants , repeatation being allowed , is
a) 105² b) 210 x 243 c) 105 x 243 d) none
76) The number of ways in which 6 different balls can be put in two boxes of different sizes so that no box remains empty is
a) 62 b) 64 c)!36 d) none
77) A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are five places in a row in his showcase. The number of different ways of displaying the 3 varieties of perfumes in the showcase is
a) 6 b) 50 c) 150 d) none
78) The numbers of arrangement of the letters of the word BHARAT taking 3 at a time is
a) 72 b)!120 c) 14 d) none
79) The number of ways to fill each of the four cells of the table with a distinct natural numbers such that the sum of the number is 10 and the sums of the numbers placed diagnolly are equal is
a) 2!2! b) 4! c) 2. 4! d) none
80) in the figure, 4 digit numbers are to be formed by filling the places with digits . The number of different ways in which the places can be filled by digits so that the sum of the numbers forme is also a 4 digit number and in no place the addition will carrying is
a) 555⁴ b) 220 c) 45⁴ d) none
81) The number of positive integral solutions of x+ y+ z= n n belongs to N, n≥ 3, is
a) ⁿ⁻¹C₂ b) ⁿ⁻¹P₂ c) n(n -1) d) none
82) The number of non negative integral solution of a+ b+ c+ d= n, n belongs to N, is
a) ⁿ⁺³P₂ b) {(n+1)(n+2)(n+3)}/6 c) ⁿ⁻¹Cₙ₋₄ d) none
83) The number of points (x,y, z)) in space, whose each co-ordinate is a negative integer such that x+ y+ z+12=0, is
a) 385 bb) 55 c) 110 d) none
84) If a,b,c are 3 natural numbers in AP and a+ b+ c= 21, then the possible number of values of the order ped triplet (a, b, c) is
a) 15 b) 14 c) 13 d) none
85) if a,b,c,,d are odd natural numbers that a + b + C + d = 20 then the number of values of the ordered quadruplet (a,b,c,d) is
a) 165 b) 455 c) 310 d) none
86) if x, y, z are integers and x≥0, y≥1, z≥2, x+ y+ z= 15 then the number of values of the ordered triplet (x,y,z) is
a) 91 b)) 455 c) ¹⁷C₁₅ d) none
87) If a,b,c are positive integer such that a + b + c≤ 8 then the number of possible values of the ordered triplet (a,b,c) is
a) 84 b) 56 c)) 83 d) none
88) The number of different ways of the distributing 10 marks among 3 questions, each question carrying atleast one mark is
a) 72 b) 71 c) 36 d) none
89) The number of ways to give away 20 apples in 3 boys, each boy receiving at least 4 apples, is
a) ¹⁰C₈ b) 90 c) ²²C₂₀ d) none
90) The position vector of a point P is r= xi + yj+ zk, where x ,y, z belongs to N and a= i+ j+ k, if r. a = 10, the number of possible position of P is
a) 36 b) 72 c) 66 d) none
**
91) If P= n(n²-1)²(n²-2²)(n²-3²).....(n²- r²), n> r, n belongs to N, then P is divisible by
a) (2r+2)! b) (2r -1)! c) (2r+1)! d) none
92) ⁿ⁺⁵Oₙ₊₁ = 11(n -1)/2. ⁿ⁺³Pₙ then the value of n is
a) 7 b) 8 c) 6 d) 9
93) If ⁿC₄, ⁿC₅ and ⁿC₆ are in AP then n is a
a) 8 b) 9 c) 14 d) 7
94) The product of r consecutive integers is divisible by
a) r b) ʳ⁻¹ₖ₌₁∑ k c) r! d) none
95) There are 10 bags B₁, B₂, B₃, .....B₁₀, which contain 21, 22, ....30 different articles respectively. The total number of ways to bring out 10 articles from a bag is
a) ³¹C₂₀ - ²¹C₁₀ b) ³¹C₂₁ c) ³¹C₂₀ d) none
96) If the number of arrangements of n-1 things taken from n different things is k times the number of arrangements of n-1 things taken from n things in which two things are identical then the value of k is
a) 1/2 b) 2 c) 4 d) none
97) Kanchana has 10 friends among whom two are married to each other. she wishes to invite five of them for a party. if the married couple refuse to attend separately then the number of different ways in which she can invite 5 friends is
a) ⁸C₅ b) 2x ⁸C₃ c) ¹⁰C₅ - 2 x ⁵C₄ d) none
98) In a plane there are two families of lines y= x+ r, y= - z+ r, where r belongs {0, 1, 2, 3, 4}. The number of squares of the diagonals of the length 2 formed by the lines is
a) 9 b) 16 c) 25 d) none
99) there are n seats round a table numbered 1, 2, 3,....n. The number ways in which m(≤ n) persons can take seats is
a) ⁿPₘ b) ⁿCₘ x (m -1)! c) ⁿ⁻¹Pₘ₋₁ d) ⁿCₘ x m!
100) Let a= i+ j + k and let r= r be a variable vector such that r.j, r.j and r.k are positive integers. If r.a≤ 12 then the number of values of r is
a) ¹²C₉ -1 b) ¹²C₃ c) ¹²C₉ d) none
101) The total number of ways in which a beggar can be given at least one rupee from four 25 paise coins, three 50 paise coins and 2 one rupee coins, is
a) 54 b) 53 c) 51 d) none
102) for the equation x+ y+ z+ w= 19, the number of positive integral solutions is equals to
a) the number of ways in which 15 identical things can be distributed among 4 persons
b) the number of ways in which 19 identical things can be distributed among four persons
c) coefficient of x¹⁹ in (x⁰+ x¹+ x²+....+ x¹⁹)⁴
d) coefficient of x¹⁹ in (x + x²+ x³+....+ x¹⁹)⁴
1b 2a 3c 4b 5a 6d 7c 8a 9b 10d 11d 12a 13c 14c 15b 16a 17b 18c 19b 20a 21b 22c 23a 24b 25b 26c 27a 28c 29b 30d 31c 32b 33a 34a 35c 36c 37b 38a 39c 40b 41a 42c 43a 44b 45d 46c 47b 48c 49c 50b 51a 52a 53b 54a 55d 56a 57c 58b 59c 60a 61c 62a 63b 64b 65c 66b 67a 68b 69a 70b 71a. 72b 73b 74a 75c 76b 77a 78c 79a 80d 81d. 82a 83b 84b 85c 86a 87a 88b 89c 90a 91a 92bc 93ac 94cd 95abc 96a 97b 98bc 99a 100ad 101bc 102a 103ad
1) If ⁿP₄ = 6 x ⁿ⁻¹P₄ then the value of n is
a) 22 b) 24 c) 48 d) 50
2) ⁿ⁺¹P₄ : ⁿ⁻¹P₃ = 72:5 Then the value of n is
a) 8 b) 9 c) 10 d) 11
3) If ¹⁰Pᵣ = 5040 then r is equal to
a) 2 b) 3 c) 4 d) 5
4) If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁ then the value of n and r are respectively equal to
a) 2,5 b) 3,2 c) 6,3 d) 7,4
5) If ¹²Pᵣ = ¹¹P₆ + 6, ¹¹P₅ then the value of r is
a) 3 b) 4 c) 5 d) 6
6) If (n +2)!= 2550. n! Then the value of n is
a) 47 b) 48 c) 49 d) 50
7) If ¹⁰Pᵣ = ⁹P₅ + 5. ⁹P₄ then the value of r is
a) 2 b) 3 c) 4 d) 5
8) If ⁴⁻ˣP₂ = 6 then the value of x is
a) 1 b) 2 c) 3 d) none
9) If ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5 Then the value of n is
a) 4 b) 3 c) 2 d) none
10) ⁿCᵣ₊₁ + ⁿCᵣ₋₁ + 2 ⁿCᵣ is equal to
a) ⁿ⁺²Cᵣ b) ⁿ⁺¹Cᵣ₊₁ c) ⁿ⁺¹Cᵣ d) ⁿ⁺¹Cᵣ₊₁
11) If ¹⁶Cᵣ = ¹⁶Cᵣ₊₂ then ʳC₄ is equal to
a) 21 b) 27 c) 35 d) 39
12) If 3 ˣ⁺¹C₂ + ²P₂ x= 4. ˣP₂, then the value of x is
a) 2 b) 3 c) 5 d) 7
13) ²ⁿC₃ : ⁿC₂ = 44:3 then n is equal to
a) 3 b) 4 c) 5 d) 6
14) If ⁿP₅ = 60. ⁿ⁻¹P₃ then n is equal to
a) 8 b) 10 c) 12 d) 14
15) The value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is
a) 24504 b) 44502 c) 42504 d) 45042
16) If ⁿC₄ = 21. ⁿ⁾²C₃ then the value of n is
a) 6 b) 10 c) 12 d) 14
17) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀∑⁴⁰⁺ʲ₁₀₊ⱼ is equal to
a) ⁷²C₃₁ b) ²⁷C₈ c) ¹⁹C₁₁ d) ⁵¹C₂₀
18) If ²⁸C₂ᵣ : ²⁴C₂ᵣ₊₄ = 225:11 then value of r is
a) 7 b) 9 c) 14 d) none
19) If ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800:1, then the value of r is
a) 11 b) 21 c) 31 d) 41
20) If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃, then
a) n<6 b) n>7 c) n < 5 d) n> 6
21) If ⁿ⁺²C₈ : ⁿ⁻²C₄ = 171:2 then the value of n is
a) 18 b) 19 c) 20 d) 21
22) The value of ⁴⁷C₄ + ⁵ᵣ₌₁∑⁵²⁻ʳC₃ is
a) 277025 b) 275027 c) 507227 d) 270725
23) ⁿPᵣ = 504 and ⁿCᵣ = 84 then the value of n is equal to
a) 3 b) 6 c) 9 d) 12
24) The number of different algebraic expressions that can be made by combining the letters p,q,r,s and t in this order with the signs '+' and '-' taking all the letters together is
a) 21 b) 23 c) 31 d) 32
25) The number of different factors of 2160 is
a) 29 b) 39 c) 49 d) none
26) 8 different chocolates can be distributed equally between two boys in
a) 70 ways b) 35 ways c) 38 ways d) 19 ways
27) From a group of person the number of ways of selecting 5 persons is equals to that of 8 persons. The number of person in the group is
a) 29 b) 25 c) 13 d) 11
28) a man has 5 oranges and 4 mangoes. How many different selections having at least one orange is possible?
a) 25 b) 30 c) 35 d) 40
29) A man has 6 friends . The number of ways in which he can invite one or more them to his house is
a) 6! b) 6! - 1 c) 2⁶! d) none
30) A 5 digited number is divisible by 3 and it is formed by using 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which such a number can be formed is
a) 126 b) 216 c) 621 d) 261
31) all the letters of the string AEPRAB are arranged in all possible ways. The number of such arrangements in which two vowels are not adjacent to each other is
a) 220 b) 115 c) 72 d) 65
32) The number of ways in which the letters of the string ANRTIPF can be arranged so that the vowels may appear in the odd place is
a) 1230 b) 1350 c) 1440 d) 1570
33) if there are 10 persons in a gathering and if each of them shakes hand with everyone else , then the number of hand shakes that takes place in the gathering is
a) 20 b) 45 c) 2¹⁰ d) 10²
34) The number of parallelograms that can be formed from a set of 4 parallel lines intersecting another set of 3 parallel lines is
a) 21 b) 20 c) 18 d) 16
35) The number of students to be selected at a time from a group of 16 students , so that the number of selection is the greatest is
a) 16 b) 14 c) 8 d) none
36) The number of different arrangement with the letters of the word ALGEBRA so that 2A's are not together is
a) 1800 b) 2520 c) 720 d) none
37) The number of odd integer of 6 significant digits can be formed with the digit 0, 1, 4, 5, 6, 7 without repetition of the digit is
a) 96 b) 108 c) 266 d) 288
38) The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant is
a) 7! b) 7!/2 c) 5(7!)/2 d) 9(7!)/2
39) The number of triangles that can be formed with 10 points as vertices, K of them them being collinear, is
a) 3 b) 5 c) 7 d) none
40) The number of ways in which 5 '+' sign and 3 'X' sign can be arranged in a row is
a) 56 b) 65 c) 72 d) 81
41) The number of ways in a which 15 class XI students and 12 class XII student be arranged in a line so that no 2 class. XII students may occupy consecutive positions is
a) 12! x 16!/4! b) 15! x 13!/4! c) 16! x 13!/ 4! d) 15! x 16!/ 4!
42) The number of strings of 3 letters that can be formed with the letter chosen from CALCUTTA is
a) 48 b) 62 c) 96 d) 102
43) The number of permutation of the letters of the MADHUBANI where the arrangementa do not begin with M but end with I is
a) 16740 b) 17460 c) 14670 d) none
44) The number of ways in which a committee of 5 persons may be formed out of 6 men and 4 women under the condition that least one woman has to be selected necessarily is
a) 252 b) 246 c) 242 d) none
45) Given that balls of the same colour are identical, the number of ways in which 18 white balls of 19 red balls may be arranged in a row so that no two white balls may come together is
a) 180 b) 190 c) 200 d) 210
46) In an examination there are 3 multiple choice questions and each question has at 4 choices. The number of ways in which one can fail to get all answers correct is
a) 12 b) 21 c) 36 d) 63
47) The number of diagonals that can be drawn by joining the vertices of an octagon is
a) 28 b) 20 c) 18 d) 16
48) out of 6 given point 3 are collinear. The number of different straight lines that can be drawn by joining any two points from those 6 given point is
a) 12 b) 10 c) 9 d) none
49) The total number of selections of at least one red ball from 4 red balls and 3 blue balls, if the balls of the same colour are different is
a) 95 b) 105 c) 120 d) 125
50) In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can unsuccessful is
a) 265 b) 255 c) 256 d) 625
51) The number of integers greater than 50000 that can be formed by the using the digits 3, 5, 6, 6, 7 is
a) 54 b) 40 c) 32 d) none
52) The number of arrangement that can be formed from the letters of the word VIOLENT, so that the vowels may occupy only odd positions is
a) 576 b) 574 c) 572 d) none
53) in a group of 15 boy there are 7 boys-scouts . The number of ways in which 12 boys can be selected from the group so as to include at least 6 boys-scouts is
a) 125 b) 127 c) 252 d) 255
54) 15 distinct objects may be divided into three groups 4, 5 and 6 objects in
a) 230230 ways b) 320320 ways c) 360360 ways d) 630630 ways
55) The number of different ways in which 1440 can be expressed as the product of two factors is
a) 18 b) 16 c) 14 d) none
56) The number of different rectangles ( regarding every square as a rectangle as well) that are there on a chess board is
a) 1280 b) 1284 c) 1296 d) 1300
57) The number of arrangements which can be made out of the letters of the word ALGEBRA without changing the relative position of the vowels and consonant is
a) 54 b) 64 c) 70 d) 72
58) The number of factors of 420 is
a) 22 b) 23 c) 24 d) none
59) A boat has a crew of 10 man of which three can row only on one side and 2 only on the other. The number of ways the crew can be arranged in the boat is
a) 142000 b) 144000 c) 124000 d) none
60) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these point is
a) 117 b) 120 c) 122 d) 124
61) The number of 6 digited integers that can be made using the digits 3 and 4 and in which at least 2 digits are different is
a) 60 b) 61 c) 62 d) none
62) The sum of the digits in unit place of all the four digited numbers formed with the help of 2,3,4,5, taken all at time is
a) 54 b) 108 c) 84 d) none
63) The number of different ways in which 15 distinct objects may be divided into three groups of 5 objects each is
a) 216216 b) 126126 c) 216612 d) 126612
64) the number of different arrangements that can be made out of the latest of the word ALLAHABAD , such that the vowels may occupy the even positions only is
a) 70 b) 50 c) 60 d) 120
65) The number of ways in which 4 letters can be posted in 3 post boxes is
a) 256 b) 81 c) 12 d) none
66) At an election, a voter may vote for any number of candidates, not greater than the number to be elected p. There are 10 candidates and 4 are to be elected. if a water votes for at least one candidate , then the number of ways in which he can vote is
a) 385 b) 1110 c) 5040 d) 6210
67) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetic order?
a) 360 b) 240 c) 120 d) 480
68) The numbers of different solutions (x,y,z) of the equation x+ y + z= 10, where each of x, y, z is a +ve integer, is
a) 36 b) ¹⁰C₃ - ¹⁰C₂ c) 10³ -10 d) none
69) If ²ⁿC₁ + ²ⁿC₂ +....+ ²ⁿCₙ₋₁ + (1/2) ²ⁿCₙ = 127, then n is equal to
a) 4 b) 5 c) 3 d) none
1b 2a 3c 4b 5d 6c 7d 8a 9a 10d 11c 12b 13d 14b 15c 16b 17d 18a 19d 20b 21b 22d 23b 24d 25b 26a 27c 28a 29d 30b 31c 32c 33b 34c 35c 36a 37d 38c 39b 40a 41d 42c 43d 44b 45b 46d 47b 48d 49c 50c 51d 52a 53c 54d 55a 56c 57d 58b 59b 60a 61c 62d 63b 64c 65b 66a 67a 68b 69a 70a
1) The sum of all the numbers of four different digits that can be made by the using the digits 0, 1, 2 and 3 is
a) 26664 b) 39996 c) 38664 d) none
2) The sum of all four digit numbers that can be formed by using the digit 2, 4, 6, 8( when repetition of digits is not allowed) is
a) 133320 b) 533280 c) 53328 d) none
3) The number of ordered pairs of integers (x,y) satisfing the equation x²+ 6x + y²= 4 is
a) 2 b) 8 c) 6 d) none
4) Among 10 persons , A, B, C are to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is
a) 10!/24 b) 9!/6 c) 10!/6 d) none
5) In how many ways can a team of 11 players be formed out of 25 players, if six out of them are always to be included and 5 always to be excluded
a) 2020 b) 2002 c) 2008 d) 8002
6) The number of ways in which the letters of the word PERSON can be placed in the squares of the given figure so that no rows remains empty is
a) 24. 6! b) 26. 6! c) 26 . 7! d) none
7) The number of words of 4 letters that can be formed from the letters of the word EXAMINATION is
a) 1464 b) 2454 c) 1678 d) none
8) The number of even divisors of the number N= 12600= ³3²5²7 is
a) 72 b) 54 cc) 18 d) none
9) In an election, the number of candidates is one greater than the persons to be elected. If a voter can vote in 254 ways , the number of candidates is
a) 7 b) 10 c) 8 d) 6
10) A person predicts the outcome of 20 cricket matches of his home team. Each match can result in a either win, loss or tie for the home team . Total number of ways in which he can make the predictions so that exactly 10 predictions are correct is equals to
a) ²⁰C₁₀ x2¹⁰ b) ²⁰C₁₀x 3²⁰ c) ²⁰C₁₀x 3¹⁰ d) ²⁰C₁₀x 2²⁰
11) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points
a) 116 b) 120 c) 117 d) none
12) There are three coplaner planet parallel lines. if any p points are taken on each of the lines, the maximum number of triangles with vertices on these points is
a) 3p²(p -1) +1
b) 3p²(p -1)
c) p²(4p -3) d) none
13) The maximum number of points of intersection of the five lines and four circles is
a) 60 b) 72 c) 62 d) none
14) The number of integral solutions of x + y + z= 0 with x≥ 5, y≥-5, z≥ -5 is
a) 134 b) 136 c) 138 d) 140
15) The number of ways in which 12 books can be put in three shelves with four on each shelf is
a) 12!/(4!)³ b) 12!/(3!)(4!)³ c) 12!/(3!)³(4!) d) none
16) Number of ways in which 25 identical things be distributed among 5 persons if each gapets odd numbers of things is
a) ²⁵C₄ b) ¹²C₅ c) ¹⁴C₁₀ d) ¹²C₃ e) none
17) The total number of divisor of 480, that are of the form 4n +2, n≥ 0, is equals to
a) 2 b) 3 c) 4
18) The number of 3 digit numbers of the form xyz such that x< y and z≤ y is
a) 276 b) 285 c) 240 d) 244
19) A man has three friends. The number of ways he can invite one friend everyday for dinner on 6 successive nights so that no friends is invited more than three times is
a) 640 b) 320 c) 420 d) 510
20) A bag contains four one rupee coins, two twenty five paise coins and five ten paise coins. In how many ways can an amount, not less than Rs 1 be taken out from the bag ? (consider coins of the same denomination to be identical)
a) 71 bb) 72 c) 73 d) 80
1c 2a 3b 4c 5b 6b 7b 8b 9c 10a 11c 12c 13c 14b 15a 16c 17c 18a 19d 20c
1) There are 10 points in a plane of which no three points are collinear but 4 points are concyclic. The number of different circles that can be drawn through atleast 3 points of these points are
a) 116 b) 120 c) 117 d) none
2) In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
a) 255 b) 256 c) 193 d) 319
3) Let 1≤ m < n ≤ p. The number of subsets of the set A={1,2,3,....p) having m,n as the least and the greatest elements respectively, is
a) 2ⁿ⁻ᵐ⁻¹ -1 b) 2ⁿ⁻ᵐ⁻¹ c) 2ⁿ⁻ᵐ d) none
4) The number of even proper divisors of 1008 is
a) 121110 d) none
5) If a,b,C be three natural numbers in AP and a+ b+ c = 21, then the possible number of values of a,b,c is
a) 15 b) 14 c) 13 d) 16
6) The number of selections of four letters from the letters of the word ASSASSINATION is
a) 72 b) 71 c) 66 d) 52
7) The number of times the digits 5 will written while listing the integers from 1 to 1000 is
a) 271 b) 275 c) 285 d) 300
8)
1) In how many ways can clean & clouded (overcast) days occur in a week assuming that an entire day is either clean or clouded.
2) Four visitors A, B C , D arrive at a town which has 5 hotels. In how many ways can they disperse themselves among 5 hotels, if 4 hotels are used to accommodate them.
3) If the letters of the word VARUN are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is
a) 98 b) 99 c) 100 d) 101
4) How many natural numbers are their 1 to 1000 which have none of their digits repeated.
5) 3 different railway passes are allotted to 5 students. The number of ways this can be done is
a) 60 b) 20 c) 15 d) 10
6) There are 6 roads between A& B and 4 roads between B and C.
a) In how many ways can one drive from A to C by way of B?
b) In how many ways can one drive from A to C and back to A, passing through B on both trips?
c) In how many ways can one drive the circular trip described in (b) without using the same road more than once.
7)a) How many car number plates can be made if each plate contains 2 different letters of English alphabet, followed by 3 different digits.
b) Solve the problem, if the first digit can not be 0.
8) a) Find the number of four letters word that can be formed from the letters of the word HISTORY. (each letter to be used atmost once)
b) How many of them contains only consonants?
c) How many of them begin with a vowel?
d) How many contains the letters Y ?
e) How many begin with T and end in a vowel?
f) How many begin with T and also contain S ?
g) How many contain both vowels?
9) If repetition are not allowed
a) How many 3 digit numbers can be formed from the six digits 2,3,5,6,7, 9
b) How many of these are less than 400?
c) How many are even?
d) How many are odd ?
e) How many are multiples of 5?
10) How many two digit numbers are there in which the tens digit and the unit digit are different are odd ?
11) Every telephone number consists of 7 digits. How many telephone numbers are there which do not include any other digits but 2,3,5, 7
12)a) In how many ways can four passengers be accommodated in three railway carriages, each carriage can accommodate any number of passengers.
b) In how many ways four persons can be accommodated in different chairs if each person can occupy only one chair.
13) How many odd numbers of five distinct digits can be formed with the digits 0,1,2,3,4?
14) Number of natural numbers between 100 and 100 such that atleast one of their digits is 7, is
a) 225 b) 243 c) 252 d) none
15) How many four digit numbers are there which are divisible by 2?
16) The 120 permutation of MAHES are arranged in dictionary order, as if each were an ordinary five letters word. The last letter of the 86th word in the list is
a) A b) H c) S d) E
17) Find the number of 7 lettered palindromes which can be formed using the letters from the English alphabets.
a)
18) Number of ways in which 7 different colours in a rainbow can be arranged if green is always in the middle.
19) Number of 4 digit numbers of the form N= abcd which satisfy three conditions:
i) 4000≤ N<6000 b) N is multiple of 5 iii) 3≤ b < c ≤ 6 is equal to
a) 12 b) 18 c) 24 d) 48
20) Find the number of ways in which the letters of the word MIRACLE can be arranged if vowels always occupy the odd places.
21) The number of 10 digit numbers such that the product of any two consecutive digits in the number is a prime, is
a) 1024 b) 2048 c) 512 d) 64
22) A letter lock consist of three rings each marked with 10 different letters. Find the number of ways in which it is possible to make an unsuccessful attempts to open the lock.
23) How many 10 digits numbers can be made with odd digits so that no two consecutive digits are same.
24) How many natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways.
¹) 128 2)120 3c 4) 738 5a 6) 124,576,360 7) 46800,421200 8) 840,120,400,240,480,40,60,240 9) 120,40,40,80,20 10) 20 11)4⁷ 12) 3⁴, 24 13) 36 14c 15 4500 16d 17)26⁴ 18)720 19c 20)5762 21b 22)299 23) 5.4⁹ 24) infinitely many
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